Question : If $\frac{x-x\tan^{2}30^{\circ}}{1+\tan^{2}30^{\circ}}=\sin^{2}30^{\circ}+4\cot^{2}45^{\circ}-\sec^{2}60^{\circ}$, then value of $x$ is:

Option 1: $\frac{1}{4}$

Option 2: $\frac{1}{5}$

Option 3: $\frac{1}{2}$

Option 4: $\frac{1}{\sqrt3}$

Question : If $\mathrm{A}=\cot 30^{\circ} \tan 60^{\circ}+\cot 60^{\circ} \tan 30^{\circ}$, then what is the value of A?

Option 1: $\frac{1}{ \sqrt{ 3}}$

Option 2: $\frac{10}{3}$

Option 3: $\frac{10}{ \sqrt{3}}$

Option 4: $\frac{1}{3}$

Question : If $A=60^{\circ}$ and $B=30^{\circ}$, find the value of $\frac{(\tan A-\tan B)}{(1+\tan A \tan B)}$.

Option 1: $3$

Option 2: $\frac{1}{\sqrt{3}}$

Option 3: $\sqrt{3}$

Option 4: $1$

Question : If $\frac{\cos \theta}{(1+\sin \theta)}+\frac{\cos \theta}{(1-\sin \theta)}=4$ and $\theta$ is acute, then the value of $\theta$ is:

Option 1: $60^{\circ}$

Option 2: $15^{\circ}$

Option 3: $45^{\circ}$

Option 4: $30^{\circ}$

Question : If $A=30^{\circ}$, then find the value of $\frac{(2 \tan A)}{\left(1-\tan^2 A\right)}$.

Option 1: $4 \sqrt{3}$

Option 2: $\frac{3}{\sqrt{3}}$

Option 3: $3$

Option 4: $2 \sqrt{3}$